Bekenstein and Mukhanov ( BM ) have suggested that , in a quantum theory of gravity , black holes may have discrete emission spectra .
Using the time-energy uncertainty principle they have also shown that , for a ( non-rotating ) Schwarzschild black hole , the natural broadening \delta \omega of the black-hole emission lines is expected to be small on the scale set by the characteristic frequency spacing \Delta \omega of the spectral lines : \zeta ^ { \text { Sch } } \equiv \delta \omega / \Delta \omega \ll 1 .
BM have therefore concluded that the expected discrete emission lines of the quantized Schwarzschild black hole are unlikely to overlap .
In this paper we calculate the characteristic dimensionless ratio \zeta ( \bar { a } ) \equiv \delta \omega / \Delta \omega for the predicted BM emission spectra of rapidly- rotating Kerr black holes ( here \bar { a } \equiv J / M ^ { 2 } is the dimensionless angular momentum of the black hole ) .
It is shown that \zeta ( \bar { a } ) is an increasing function of the black-hole angular momentum .
In particular , we find that the quantum emission lines of Kerr black holes in the regime \bar { a } \gtrsim 0.9 are characterized by the dimensionless ratio \zeta ( \bar { a } ) \gtrsim 1 and are therefore effectively blended together .
Our results thus suggest that , even if the underlying mass ( energy ) spectrum of these rapidly-rotating Kerr black holes is fundamentally discrete as suggested by Bekenstein and Mukhanov , the natural broadening phenomenon ( associated with the time-energy uncertainty principle ) is expected to smear the black-hole radiation spectrum into a continuum .